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%{\LARGE\bf 上海立信会计金融学院期终考试卷 --- 试题纸} \hspace{0.3cm} {\Large \underline{ A }卷 }
{\Large\bf \H 上海立信会计金融学院期终考试卷 } \hspace{0.3cm} {\Large \underline{ B }卷 }

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{\large \bf \H 2023 $\sim$ 2024 学年 第 二 学期 }

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{\large \bf \H \underline{ \emph{2021级数学与应用数学专业} } 《\underline{ \emph{多元统计分析} }》 课程代码：\underline{ 160290220 }  }

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{\H（本场考试属\underline{  开  }卷考试，考试时间共\underline{  90  }分钟，不准使用计算器）共\underline{  4  }页 }
%{\large （本场考试属闭卷考试，考试时间 90 分钟，禁止使用计算器） }

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%{\large 本考试卷共 4 页，请在本考试卷上答题。}

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班级 \underline{\hspace{3.5cm}} 学号 \underline{\hspace{3.5cm}} 姓名 \underline{\hspace{3.5cm}} 

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题号 &一&二&三&四&五&六&七&八&九&十&总分&合成人签名&审核人签名 \\
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%应得分&15&15&15&15&15&15&10&100 \\
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本次考试共10题，每题10分。
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\item %1 第二章：随机向量：第46页：习题2.9 
设随机向量 $x=(x_1,x_2,\cdots,x_p)^t$ 的均值向量和协方差矩阵分别为 $\mu$ 和 $\Sigma$. 
证明 $E(xx^t) = \Sigma + \mu\mu^t$.  

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\item %2 第三章：多元正态分布：3.2. 多元正态分布的性质：第52页：例子3.2.3. 
设 $x=(x_1,x_2)^t \sim N_2(\mu, \Sigma)$,  设 $a=(a_1,a_2)^t$ 是常数向量，求 $y=a^tx$ 的分布。

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\item %3 第三章：多元正态分布：3.4. 复相关系数和偏相关系数：第63页：最优线性预测
用随机向量 $x$ 的函数 $g(x)$ 来预测随机变量 $y$ 时，可用均方误差 $E[y-g(x)]^2$ 
作为预测精度的度量。如果限制 $g(x)$ 为线性函数，则使得均方误差达到最小的线性预测函数是什么？ 
最优线性预测的均方误差是什么？

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\item %4 第三章：多元正态分布：3.5. 样本均值和样本协方差矩阵的抽样分布：第69页
写出自由度为 $n$ 的 $p$ 阶 Wishart 分布的定义。并解释 Wishart 分布与 $\chi^2$ 分布的关系。

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\item %5 第四章：多元正态总体的统计推断：4.3. 两个总体均值的比较推断：第98页：成对试验的T^2统计量
设 $(x_i,y_i), i=1,2,\cdots,n$ 是成对试验的数据，令 $d_i=x_i-y_i$, 又设 $d_1,d_2,\cdots,d_n$ 独立同分布于 $N_p(\delta,\Sigma)$, 其中 $\Sigma>0$, $\delta=\mu_1-\mu_2$ 是总体 $x$ 和 $y$ 的均值向量的差。
考虑假设检验 $$H_0: \delta=0, \,\,v.s.\,\, H_1:\delta\neq 0. $$
写出检验统计量和拒绝规则。 

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\item %6 第四章：多元正态总体的统计推断：第112页：4.7.总体相关系数的推断
设多元正态总体 $\vec{x}\sim N_p(\mu,\Sigma)$, $\Sigma>0$, 设 $\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n$ 是从总体 $x$ 中抽取的一个简单随机样本。设 $\vec{x}=(x_1,\cdots,x_p)^t$. 记 $\rho_{ij}=\rho(x_i,x_j)$ 是总体相关系数。考虑假设检验
$$H_0: \rho_{ij}=0, \,\,\mathrm{vs.}\,\, H_1: \rho_{ij}\neq 0. $$
写出检验统计量和拒绝规则。

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\item %7 第五章：判别分析：第172页：习题5.2
设对来自组 $\pi_1$ 和组 $\pi_2$ 的两个样本有 
$$
\bar{x}_1=\begin{pmatrix} 5 \\ 3 \end{pmatrix}, \,\,\, 
\bar{x}_2=\begin{pmatrix} 3 \\ 1 \end{pmatrix}, \,\,\, 
S_p=\begin{pmatrix} 5 & 2 \\ 2 & 4 \end{pmatrix}, 
$$
设 $\Sigma_1=\Sigma_2$, 试给出距离判别规则，并将 $x_0=(3,3)^t$ 分到组 $\pi_1$ 或 $\pi_2$. 

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\item %8 第六章：聚类分析：6.3. 系统聚类法：第185页：例子6.3.2.
设有四个样品，每个样品只测量了一个指标，分别是 $1,2,6,8$. 
定义类与类之间的距离为所有样品对之间的平均距离，距离定义为指标的差的绝对值。
使用类平均法进行聚类。写出每次聚类得到的距离矩阵。

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\item %9 第七章：主成分分析：7.3.样本的主成分：第226页：一、样本主成分的定义
设 $x=(x_1,\cdots,x_p)^t$ 是一个 $p$ 维随机向量。
设 $\vec{x}_1,\cdots,\vec{x}_n$ 是一个样本，其中每个 $\vec{x}_i$ 都是一个 $p$ 维列向量。
%记 $X=(\vec{x}_1,\cdots, \vec{x}_n)^t = (x_{ij})_{n\times p}$ 为数据矩阵。
写出第一样本主成分和第二样本主成分的计算思路。

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\item %10 第八章：因子分析：8.2. 正交因子模型：第255页：三、因子载荷矩阵的统计意义
考虑正交因子模型 $x=\mu +Af+\varepsilon$ 的因子载荷矩阵 $A$ 的统计意义。
给出矩阵 $A$ 的元素的含义、行元素平方和的含义、列元素平方和的含义、以及所有元素的平方和的含义。

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